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Maths-

General

Easy

Question

If the curve $y equals a x to the power of 3 end exponent plus b x to the power of 2 end exponent plus c x$ is inclined at $45 degree$ to $x$ -axis at (0, 0) but touches $x$ -axis at (1, 0), then

$a = 1, b = - 2, c = 1$
$a = 1, b = 1, c = - 2$
$a = - 2, b = 1, c = 1$
$a = - 1, b = 2, c = 1$

The correct answer is: $a equals 1 comma blank b equals negative 2 comma blank c equals 1$

We have,
$y equals a x to the power of 3 end exponent plus b x to the power of 2 end exponent plus c x$ …(i)

$rightwards double arrow fraction numerator d y over denominator d x end fraction equals 3 a x to the power of 2 end exponent plus 2 b x plus c$ …(ii)

It is given that

$open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis 0 comma blank 0 right parenthesis end subscript equals tan invisible function application 45 degree rightwards double arrow c equals 1$

Also,

$open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis 1 comma blank 0 right parenthesis end subscript equals 0 rightwards double arrow 3 a plus 2 b plus c equals 0 rightwards double arrow 3 a plus 2 b plus 1 equals 0 blank open square brackets because c equals 1 close square brackets$

Clearly, $a equals 1$ and $b equals negative 2$ satisfy this equation

Hence, $a equals 1 comma blank b equals negative 2$ and $c equals 1$

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